Problem: Alicia watched a drone take off from a bridge. The height of the drone (in meters above the ground) $t$ minutes after takeoff is modeled by $h(t)=-3t^2+12t+96$ Alicia wants to know when the drone will land on the ground. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. $h(t)=$ 2) How many minutes after takeoff does the drone land on the ground?
Explanation: Choosing a form When the drone lands on the ground, its height above the ground is $0$ meters. So we're looking for what values of $t$ make the output of the function $0$. Which form reveals this feature? Here's a summary of what each form reveals along with examples. Note that these are all equivalent forms of the same function, but not the function modeling the height of the drone. Form Example Feature revealed Standard $f(x)=2x^2-12x+{10}$ $y$ -intercept is ${10}$ Factored $f(x)=2(x-C{1})(x-C{5})$ Zeros are $x=C1$ and $x=C5$ Vertex $f(x)=2(x-{3})^2{-8}$ Vertex is $(3,{-8})$ Rewrite in factored form The zeros of the function tell us which values of $t$ make the output of the function $0$, so let's rewrite $h(t)$ in factored form: $\begin{aligned} h(t)&=-3t^2+12t+96 \\\\ &=-3(t^2-4t-32) \\\\ &=-3\left(t+4\right)\left(t-8\right) \end{aligned}$ When does the drone land on the ground? The factored form of the function reveals its zeros: $\begin{aligned} &0=-3\left(t+4\right)\left(t-8\right) \\\\ &t+4=0\text{ or }t-8=0 \\\\ &\xcancel{t=-4} \text{ or }t=8 \end{aligned}$ A negative value for time doesn't make sense in this context, so the drone lands on the ground at $t=8$ minutes. Answers 1) The factored form of the function reveals when the drone lands on the ground: $h(t)=-3\left(t+4\right)\left(t-8\right)$ 2) The drone lands on the ground $8$ minutes after takeoff.